Hi, Firstly, it would be "to scramble one's eggs" which literally means to cook my eggs. Sometimes in English we use the word 'one' to mean my or mine. It's usually used in very upper class English, such as the Royal Family. The Queen may say "Please scramble one's eggs this morning, rather than fry them". It can also mean to wind someone up, tease them and get them worked up.

The correct version of this phrase would be "to scramble one's eggs", since it is not possible to have an infinitive in past participle form. Scrambled eggs are eggs cooked with the yellow and white parts mixed together. So, to scramble one's eggs literally refers to the process of scrambling (mixing) the eggs while cooking them.